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On Contra – RW Continuous Functions in Topolo...
On Contra – RW Continuous Functions in Topological Spaces
Example 3.2: Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a...
On Contra – RW Continuous Functions in Topological Spaces
Proof: Let f: X → Y be contra rw-continuous. Let F be any closed...
On Contra – RW Continuous Functions in Topological Spaces
Example 3.31: In example 3.29, f is rw-irresolute but not perfec...
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On Contra – RW Continuous Functions in Topological Spaces

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On Contra – RW Continuous Functions in Topological Spaces

  1. 1. International OPEN Journal ACCESS Of Modern Engineering Research (IJMER) On Contra – RW Continuous Functions in Topological Spaces M. Karpagadevi1, A. Pushpalatha2 1 Assistant Professor, S.V.S. College of Engineering, Coimbatore, India 2 Assistant Professor, Government Arts College, Udumalpet, India ABSTRACT: In this paper we introduce and investigate some classes of functions called contra-rw continuous functions. We get several characterizations and some of their properties. Also we investigate its relationship with other types of functions. Mathematics Subject Classification: 54C08, 54C10 Keywords: Contra rw-continuous, Almost Contra rw-continuous, Contra rw-irresolute I. Introduction In 1996, Dontchev[1] presented a new notion of continuous function called contra-continuity. This notion is a stronger form of LC-Continuity. In 2007, Benchalli.S.S. and Wali.R.S.[2] introduced RW-closed set in topological spaces. In 2012, Karpagadevi.M. and Pushpalatha.A.[3] introduced RW-continuous maps and RW-irresolute maps in topological spaces. The purpose this paper is to define a new class of continuous functions called Contra-rw continuous functions and almost Contra rw-Continuous function and investigate their relationships to other functions. II. Preliminaries Throughout this paper X and Y denote the topological spaces (X,) and (Y,σ) respectively and on which no separation axioms are assumed unless otherwise explicitly stated. For any subset A of a space (X,), the closure of A, the interior of A and the complement of A are denoted by cl(A), int(A) and Ac respectively. (X,) will be replaced by X if there is no chance of confusion. Let us recall the following definitions as pre requesters. Definition 2.1: A Subset A of (X,) is called (i) generalized closed set (briefly g-closed) [5] if cl(A)  U whenever A  U and U is open in X. (ii) regular generalized closed set (briefly rg-closed)[6] if cl(A)  U whenever A  U and U is regular open in X. The Complement of the above mentioned closed sets are their respective open sets. Definition 2.2: A Subset A of a Space X is called rw-closed [2] if cl(A)  U whenever A  U and U is regular semiopen in X. Definiton 2.3: A Map f: (X,) → (Y,σ) is said to be (i) rw-continuous[3] if f-1(V) is rw-closed in (X,) for every closed set V in (Y,σ) (ii) rw-irresolute[3] if f-1(V) is rw-closed in (X,) for every rw-closed set V of (Y,σ) (iii) rw-closed[2] if f(F) is rw-closed in (Y,σ) for every rw-closed set F of (X,) (iv) rw-open[2] if f(F) is rw-open in (Y,σ) for every rw-open set F of (X,) Definiton 2.4: A Map f: (X,) → (Y,σ) is said to be contra-continuous [1] if f-1 (V) is closed in (X,) for every open set V in (Y,σ). III. Contra RW-Continuous Function In this section we introduce the notions of contra rw-continuous, contra rw-irresolute and almost contra rw-continuous functions in topological spaces and study some of their properties. Definition 3.1: A function f: (X,) → (Y,σ) is called contra rw-continuous if f-1 (V) is rw-closed set in X for each open set V in Y. | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 2 | Feb. 2014 |13|
  2. 2. On Contra – RW Continuous Functions in Topological Spaces Example 3.2: Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{a,b}}. Let f: X → Y be a map defined by f (a) = a, f (b) = b and f(c) = c. Clearly f is contra rw-continuous. Theorem 3.3: Every contra-continuous function is contra rw-continuous. Proof: The proof follows from the fact that every closed set is rw-closed set. Remark 3.4: The converse of the above theorem need not be true as seen from the following example. Example 3.5: Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{a,b}}. Let f: X → Y be a map defined by f (a) = a, f (b) = b and f(c) = c. Clearly f is contra rw-continuous but not contra-continuous since f -1({a,b}) = {a,b} which is not closed in X. Theorem 3.6: If a function f: X → Y is contra rw-continuous, then f is contra-continuous. Proof: Let V be an open set in Y. Since f is contra rw-continuous, f -1(V) is closed in X. Hence f is contracontinuous. Remark 3.7: The concept of rw-continuity and contra rw-continuity is independent as shown in the following examples. Example 3.8 : Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{b,c}}. Let f: X → Y be a map defined by f (a) = a, f (b) = b and f(c) = c. Clearly f is contra rw-continuous but not rw-continuous since f -1({a}) = {a} is not rw-closed in X where {a} is closed in Y. Example 3.9 : Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{b}}. Define f: X → Y identity mapping. Then c learly f is rw-continuous but not contra rw-continuous since f -1({b}) = {b} is not rw-closed in X where {b} is closed in Y. Theorem 3.10: Every contra rw-continuous function is contra rg-continuous. Proof: Since every rw-closed set is rg-closed, the proof is obvious. Remark 3.11: The converse of the above theorem need not be true as seen from the following example. Example 3.12: Let X=Y={a,b,c,d} with topologies ={X,,{a},{b},{a,b},{a,b,c}} and σ = {Y,,{a},{b},{a,b,c}}. Let f: X → Y be a map defined by f (a) = c, f (b) = d, f(c) = a and f (d) =b. Clearly f is contra rg-continuous but not contra rw-continuous since f -1({a}) = {c} is not rw-closed in X where {a} is open in Y. Remark 3.13: The composition of two contra rw-continuous functions need not be contra rw-continuous as seen from the following example. Example 3.14: Let X=Y=Z={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{a,b}} and  = {Z, ,{a}}. Let f: X → Y be a map defined by f (a) = c, f (b) = b and f(c) = a and g: Y→Z is defined by g (a) =b, g (b) =c and g(c) =a.Then clearly f and g are contra rw-continuous. But g  f: X → Z is not contra rw-continuous since (g  f)-1{a} = f-1(g-1{a}) = f-1 ({c}) = {a} is not rw-closed in X. Theorem 3.15: If f: (X,) → (Y,σ) is contra rw-continuous and g: Y→Z is a continuous function, then g  f : X→Z is contra rw-continuous. Proof: Let V be open in Z. Since g is continuous, g -1(V) is open in Y. Then f -1(g -1(V)) is rw-closed in X since f is contra rw-continuous. Thus (g  f) -1 (V) is rw-closed in X. Hence g  f is contra rw-continuous. Theorem 3.16: If f: X → Y is rw-irresolute and g: Y→Z is contra continuous function then g  f: X→Z contra rw-continuous. Proof: Since every contra continuous is contra rw-continuous, the proof is obvious. is Theorem 3.17: If f: X → Y is contra rw-continuous then for every x X, each F C(Y,f(x)), there exists URWO(X,x) such that f(U)  F (ie) For each x X, each closed subset F of Y with f(x)  F, there exists a rw-open set U of Y such that xU and f(U)F. | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 2 | Feb. 2014 |14|
  3. 3. On Contra – RW Continuous Functions in Topological Spaces Proof: Let f: X → Y be contra rw-continuous. Let F be any closed set of Y and f(x)  F where xX. Then f -1(F) is rw-open in X. Also x  f -1(F). Take U = f -1(F). Then U is a rw-open set containing x and f (U)  F. Theorem 3.18: Let (X,) be a rw-connected space (Y,σ) be any topological space. If X → Y is surjective and contra rw-continuous then Y is not a discrete space. Proof: Suppose Y is discrete space. Let A be any proper non-empty subset of Y. Then A is both open and closed in Y. Since f is contra rw-continuous, f -1(A) is both rw-open and rw-closed in X. Since X is rw-connected, the only subset of Y which are both rw-open and rw-closed are X and. Hence f -1(A) = X. Then it contradicts to the fact that f: X → Y surjective. Hence Y is not a discrete space. Definition 3.19: A function f: X → Y is called almost contra rw-continuous if f -1(V) is rw-closed set in X for every regular open set V in Y. Example 3.20: Let X=Y=Z={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ ={ Y,,{a},{c},{a,c}}. Define f: X → Y by f (a) =a, f (b) =b and f(c)=c. Here f is almost contra rw-continuous since f -1({a}) = {a} is rw-closed in X for every regular open set {a} in Y. Theorem 3.21: Every contra rw-continuous function is almost contra rw-continuous. Proof: The proof follows from the definition and fact that every regular open set is open. Definition 3.22: A function f: X → Y is called contra rw-irresolute if f -1(V) is rw-closed in X for each rw-open set V in Y. Definition 3.23: A function f: X → Y is called perfectly contra rw-irresolute if f -1(V) is rw-closed and rw-open in X for each rw-open set V in Y. Theorem 3.24: A function f: X → Y is perfectly contra rw-irresolute if and only if f is contra rw-irresolute and rw-irresolute. Proof: The proof directly follows from the definitions. Remark 3.25: The following examples shows that the concepts of rw-irresolute and contra rw-irresolute are independent of each other. Example 3.26: Let X=Y=Z={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ ={ Y,,{a},{c},{a,c}}. Define f: X → Y by f (a) = a, f (b) = b and f(c) = c. Here f is contra rw-irresolute but not rw-irresolute since f -1({a,b}) = {a,b} is not rw-open in X. Example 3.27: Let X=Y=Z={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ ={ Y,,{a},{c},{a,c}}. Define f: X → Y by f (a) = a, f (b) = b and f(c) = c. Here f is rw-irresolute but not contra rw-irresolute since f -1({b}) = {b} is not rw-closed in X. Theorem 3.28: Let f: X → Y and g: Y → Z be a function then (i) If g is rw-irresolute and f is contra rw-irresolute then g  f is contra rw-irresolute (ii) If g is contra rw-irresolute and f is rw-irresolute then g  f is contra rw-irresolute Proof: (i) Let U be a rw-open in Z. Since g is rw-irresolute, g -1(U) is rw-open in Y. Thus f -1(g -1(U)) is rw-closed in X. Since f is contra rw-irresolute (ie) (g  f) -1(U) is rw-closed in X. This implies that g  f is contra rw-irreolute. (ii) Let U be a rw-open in Z. Since g is contra rw-irresolute, g -1(U) is rw-closed in Y. Thus f -1(g -1(U)) is rw-closed in X since f is rw-irresolute (ie) (g  f) -1(U) is rw-closed in X. This implies that g  f is contra rw-irresolute. Theorem 3.29: Every perfectly contra rw-irresolute function is contra rw-irresolute and rw-irresolute. Proof: The proof directly follows from the definitions. Remark 3.30: The following two examples shows that a contra rw-irresolute function may not be perfectly contra rw-irresolute and rw-irresolute function may not be perfectly contra rw-irresolute. | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 2 | Feb. 2014 |15|
  4. 4. On Contra – RW Continuous Functions in Topological Spaces Example 3.31: In example 3.29, f is rw-irresolute but not perfectly contra rw-irresolute and in example 3.28, f is contra rw-irresolute but not perfectly contra rw-irresolute. Theorem 3.32: A function is perfectly contra rw-irresolute iff f is contra rw-irresolute and rw-irresolute. Proof: The proof directly follows from the definitions. REFERENCES [1]. [2]. [3]. [4]. [5]. [6]. [7]. Dontchev.J.”Conra-continuous functions and strongly s closed spaces”, International J.Math&Math.Sci.19 (1996).303-310. Benchalli.S.S. and Wali.R.S, “On RW-closed sets in topological spaces”, Bull. Malays. Math. Sci. Soc (2) 30(2) (2007), 99-110. Karpagadevi.M. and Pushpaltha. A., “RW-Continuous maps and RW-irresolute maps in topological spaces”, International Journal of Mathematics Trends and Technology, Vol.4, Issue 2-2013, 21-25. Karpagadevi.M. and Pushpaltha. A., “RW-Closed maps and RW- Open maps in topological spaces”, International Journal of Computer Applications Technology and Research, Vol.2, Issue 2-2013, 91-93. Levine.N. ”Generalized closed sets in topology”, Rend.Circ.Mat.Palermo 19.1970,89-96. Palaniappan.N and Rao.K.C. “Regular generalized closed sets”, Kyungpook Math J 33, 1993, 211-219. Syed Ali Fathima.S. and Mariasingam.M. “On contra #rg-Continuous functions”, International Journal of Modern Engineering Research, Vol.3.Issue.2, March-April.2013 pp.939-943 | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 2 | Feb. 2014 |16|

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